Algebraic Properties of Permutation Polynomials

Eiji OKAMOTO; Wayne AITKEN; George Robert BLAKLEY

Algebraic Properties of Permutation Polynomials

Eiji OKAMOTO, Wayne AITKEN, George Robert BLAKLEY

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Polynomials are called permutation polynomials if they induce bijective functions. This paper investigates algebraic properties of permutation polynomials over a finite field, especially properties associated with permutation cycles. A permutation polynomial has a simple structure but good randomness properties suitable for applications. The cycle structure of permutations are considered to be related to randomness. We investigate the algebraic structure from the viewpoint of randomness. First we show the relationship between polynomials and permutations using a matrix equation. Then, we give a general form of a permutation polynomial corresponding to a product C₁C₂・・・C_k of pairwise disjoint cycles. Finally, permutation polynomials with fixed points -or with 2, 3 and 4-cycles -and their compositions are given together with distribution of degree of the permutation polynomials.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E79-A No.4 pp.494-501

Publication Date: 1996/04/25

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Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

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Eiji OKAMOTO, Wayne AITKEN, George Robert BLAKLEY, "Algebraic Properties of Permutation Polynomials" in IEICE TRANSACTIONS on Fundamentals, vol. E79-A, no. 4, pp. 494-501, April 1996, doi: .
Abstract: Polynomials are called permutation polynomials if they induce bijective functions. This paper investigates algebraic properties of permutation polynomials over a finite field, especially properties associated with permutation cycles. A permutation polynomial has a simple structure but good randomness properties suitable for applications. The cycle structure of permutations are considered to be related to randomness. We investigate the algebraic structure from the viewpoint of randomness. First we show the relationship between polynomials and permutations using a matrix equation. Then, we give a general form of a permutation polynomial corresponding to a product C₁C₂・・・C_k of pairwise disjoint cycles. Finally, permutation polynomials with fixed points -or with 2, 3 and 4-cycles -and their compositions are given together with distribution of degree of the permutation polynomials.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/e79-a_4_494/_p

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@ARTICLE{e79-a_4_494,
author={Eiji OKAMOTO, Wayne AITKEN, George Robert BLAKLEY, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Algebraic Properties of Permutation Polynomials},
year={1996},
volume={E79-A},
number={4},
pages={494-501},
abstract={Polynomials are called permutation polynomials if they induce bijective functions. This paper investigates algebraic properties of permutation polynomials over a finite field, especially properties associated with permutation cycles. A permutation polynomial has a simple structure but good randomness properties suitable for applications. The cycle structure of permutations are considered to be related to randomness. We investigate the algebraic structure from the viewpoint of randomness. First we show the relationship between polynomials and permutations using a matrix equation. Then, we give a general form of a permutation polynomial corresponding to a product C₁C₂・・・C_k of pairwise disjoint cycles. Finally, permutation polynomials with fixed points -or with 2, 3 and 4-cycles -and their compositions are given together with distribution of degree of the permutation polynomials.},
keywords={},
doi={},
ISSN={},
month={April},}

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TY - JOUR
TI - Algebraic Properties of Permutation Polynomials
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 494
EP - 501
AU - Eiji OKAMOTO
AU - Wayne AITKEN
AU - George Robert BLAKLEY
PY - 1996
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E79-A
IS - 4
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - April 1996
AB - Polynomials are called permutation polynomials if they induce bijective functions. This paper investigates algebraic properties of permutation polynomials over a finite field, especially properties associated with permutation cycles. A permutation polynomial has a simple structure but good randomness properties suitable for applications. The cycle structure of permutations are considered to be related to randomness. We investigate the algebraic structure from the viewpoint of randomness. First we show the relationship between polynomials and permutations using a matrix equation. Then, we give a general form of a permutation polynomial corresponding to a product C₁C₂・・・C_k of pairwise disjoint cycles. Finally, permutation polynomials with fixed points -or with 2, 3 and 4-cycles -and their compositions are given together with distribution of degree of the permutation polynomials.
ER -